\magnification = 2000 
\input amstex
\documentstyle{amsppt}
\input OurATOMacros
\input OurPlainGraphicsMacros

\vglue -5pt

% File Name as ATO: Dirac Belt Trick

\Title The Dirac Belt Trick. 

\LF
Dirac invented the famous belt trick to demonstrate a property of the motions of Euclidean
Space that is indeed difficult to imagine without visual help.
\Lf
The trick is performed with a strip, or belt, that is initially parallel to the screen.  The orthogonal
projection of the performance looks as follows: The left end of the belt stays fixed, the right
end moves around the left end in a circular motion. It is important that the moving end 
{\bf stays parallel} to the fixed end through the whole trick (parallel means: the final edge
and the final normal each stay parallel to the initial edge and initial normal). One observes
with surprise: \lf
{\it After moving the right end once around the circle the belt is twisted twice. After the second
circular move the belt is untwisted (as it was initially).}
\Lf
The trick is shown in stereo because it is impossible that the belt stays in its initial plane
when the ends are moved as described. It is important to visualize how the 
different parts of the belt move {\bf vertically} to the screen.
\Lf
It will be no surprise to observe that the first circular movement -- when looked at in 3D --
is different from the second circular movement . During the first circular movement  the
middle part of the belt moves vertically to be {\bf in front} of the screen and the two ends move
vertically to be {\bf behind} the screen. During the second circular movement  it is the other
way round: the middle part of the belt moves vertically to be behind the screen and 
the ends move vertically to be in front of  the screen. As soon as one can fix this image
in one's mind it is obvious how the circular motion with parallel ends produces the 
twist of the belt.
\Lf
Another instance of the same property of the motions is the {\it waiter's cup trick} : It is possible 
to continuously rotate a cup on ones horizontal hand in the {\bf same} direction if during the 
first rotation the hand is above elbow height, during the second rotation below elbow height
and so on, alternatingly above and below elbow height. Namely, imagine that the shoulder
is the fixed end of the belt and the always horizontal (!) middle part of the belt is the hand.

\noindent
Bob Palais

\bye